Robert Silverberg(click on names to see more mathematical fiction
by the same author)

...

Contributed by
Vijay Fafat

The novel is set on a water-logged planet called “Hydros”, populated by artificial islands floating on a planet-spanning ocean. A few humans on one of the islands end up offending the local “Dwellers” and are ordered to evacuate. Banished, the humans set out on a Homeric journey to find another island and end up reaching a vast, mysterious continent called “The Face of the Waters”. A mystical ending follows.

As they approach the Face, a priest breaks out into sudden mathematical epiphany (which, unfortunately, is quite incorrect in my opinion. Silverberg appears to have confused Godel’s Theorem with the axiomatic framework of mathematics, as in the following monologue):

(quoted from The Face of the Waters)

'Let me make use of a mathematical analogy. Do you know what
Godel's Theorem is?"

'No."

'Well, neither do I, not exactly. But I can give you an
approximation of it. It's a twentieth-century idea, I think. What
Godel's Theorem asserts, and nobody has ever been able to disprove
it, is that there's a fundamental limit to the rational reach
of mathematics. We can prove all the assumptions of mathematical
reasoning down to a certain bed-rock point, and then we hit a level
where we simply can't go any farther. Ultimately we find that we've
descended beyond the process of mathematical proof to a realm of
unprovable axioms, things that simply have to be taken on faith
if we're to make any sense out of the universe. What we reach
is the boundary of reason. In order to go beyond it - in order
to go on thinking at all, really - we are compelled to accept our
defining axioms as true, even though we can't prove them. Are you
following me?"

followed by the speculation:

(quoted from The Face of the Waters)

'All right. What I propose is that Godel's Theorem marks
the dividing line between gods and mortals."

'Really," Lawler said.

'This is what I mean," said Quillan. 'It sets a boundary for reasoning. The gods occupy the far side of that boundary. Gods, by definition, are creatures who aren't bound by the Godel limits. We humans live in a world where reality ultimately breaks down into irrational assumptions, or at least assumptions that are non-rational because they're unprovable. Gods live in a realm of absolutes where realities are not only fixed and knowable down beyond the level of our axiomatic floor, but can be redefined and reshaped by divine control."

For the first time in this discussion Lawler felt a flicker of interest. 'The galaxy is full of beings which aren't human, but
their maths isn't any better than ours, is it? Where do they fit your
scheme?"

'Let's define all intelligent beings who are subject to the
Godel limitations as human, regardless of their actual species. And
any beings that are capable of functioning in an ultra-Godelian realm
of logic are gods."

And

(quoted from The Face of the Waters)

'Let us suppose that the gods themselves at some point must reach a Godel limit, a place where their own reasoning powers - that is, their powers of creation and recreation - run up against some kind of barrier. Like us, but on a qualitatively different plane, they eventually come to a point at which they can go thus far, and no farther."

'The ultimate limit of the universe," Lawler said.

'No. Just their ultimate limit. It may well be that there are greater gods beyond them. The gods we're talking about are encapsulated just as we mortals are within a larger reality defined by a different mathematics to which they have no access. They look upward to the next reality and the next level of gods. And those gods - that is, the inhabitants of that larger reality - also have a Godel wall around them, with even greater gods outside it. And so on and so on and so on."

Finally ending up with some Cantorian speculation about Absolute Infinity:

(quoted from The Face of the Waters)

Lawler felt dizzy. 'To infinity?"

'Yes."

'But don't you define a god as something that's infinite?
How can an infinite thing be smaller than infinity?"

'An infinite set may be contained within an infinite set.
An infinite set may contain an infinity of infinite subsets."

'If you say so," replied Lawler, a little restless now. 'But
what does this have to do with the Face?"

'If the Face is a true Paradise, unspoiled and virgin - a
domain of the holy spirit - then it may very well be occupied by
superior entities, beings of great purity and power. What we of the
Church once called angels. Or gods, as those of older faiths might
have said."

Be patient, Lawler thought. The man takes these things
seriously.

He said, 'And these superior beings, angels, gods, whatever
term we choose to use - these are the local post-Godelian geniuses,
do I have it right? Gods, to us. Gods to the Gillies, too, since the
Face seems to be a holy place for them. But not God Himself, God
Almighty, your god, the one that your church worships, the prime
creator of the Gillies and us and everything else in the universe.
You won't find Him around. here, at least not very often. That god
is higher up along the scale of things. He doesn't live on any one
particular planet. He's up above somewhere in a higher realm, a
larger universe, looking down, checking up occasionally on how
things are going here."

'Exactly."

'But even He isn't all the way at the top?"

'There is no top," Quillan said. 'There's only an ever-retreating
ladder of Godhood, ranging from the hardly-more-than-mortal
to the utterly unfathomable. I don't know where the
inhabitants of the Face are located on the ladder, but very likely it's
somewhere at a point higher than the one we occupy. It's the whole
ladder that is God Almighty. Because God is infinite, there can be
no one level of godhood, but only an eternally ascending chain;
there is no Highest, merely Higher and Higher and Even Higher,
ad infinitum. The Face is some intermediate level on that chain."

'I see," said Lawler uncertainly.

'And by meditating on these things, one can begin to perceive
the higher infinities, even though by definition we can never
perceive the Highest of all, since to do that we'd have to be greater
than the greatest of infinities." Quillan looked toward the heavens
and spread his arms wide in a gesture that was almost self-mocking.

Thanks to Vijay for bringing this book to my attention. Although Gödel's mathematical work appears to play only a minor part in the novel, it certainly belongs here in the Mathematical Fiction database. The idea that there are infinitely many different "Gödelian levels" is almost certainly nonsense. (It doesn't make any sense to me anyway.) However, it is interesting and has a certain familiar mathematical flavor that can be found in real mathematics. Similarly, although I agree that the description of Gödel's work in this book is not quite accurate, I am not as critical of it as Vijay. In particular, though it does not say anything about "undecidability" (one key component of the theory), the character's description comes close to capturing the idea that the consistency of finite axiomatic systems cannot be confirmed within the system itself. This does not really have anything to do with the question of whether the axioms are "true" (whatever that may mean), but it does mean that when mathematicians are working within a system they are (at least temporarily) accepting its consistency as an "article of faith".

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